ME 442 – Analytical Compensator Design in Frequency Domain Page

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Analytical Phase Lag, Phase Lead and PID Design in Frequency Domain Based on the following articles:

1. W. R. Wakeland, “Bode Compensator Design”, IEEE Trans. on Autom. Control, Vol. AC-21, October 1976, p. 771.

2. J. R. Mitchell, “Comments on Bode Compensator Design”, IEEE Trans. on Autom. Control, Vol. AC-22, October 1977, p. 869.

For a given open loop transfer function G(s)H(s) with an uncompensated gain crossover frequency ωg,u, if it is desired to design

a) a phase lag compensator

1withTs1

Ts1K)s(G cc >αα++

= (1)

b) a phase lead compensator

1withTs1

Ts1K)s(G cc <αα++

= (2)

c) a PID controller

sKs

KK)s(G di

pc ++= (3)

which will lead to a compensated closed loop system with i) a desired phase margin φm, ii) at a desired gain crossover frequency ωg,c,

then the following analytical procedure may be followed. This procedure yields answers if they actually exist. Define the angle of the controller of interest Gc(s) at the desired gain crossover frequency ωg,c as

θ = arg[Gc(jωg,c)] = –180° + φm – arg[G(jωg,c)H(jωg,c)] (4)

Then the design equations for • a lag or lead controller are

θωωω

θωω−=

sin)j(H)j(GK

cos)j(H)j(GK1T

c,gc,gcc,g

c,gc,gc (5a)

θω

ωω−θ=α

sin)j(H)j(GKcos

Tc,g

c,gc,gc (5b)

• a PID controller are

)j(H)j(GcosK

c,gc,gp ωω

θ= (6a)

ME 442 – Analytical Compensator Design in Frequency Domain Page

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)j(H)j(GsinKK

c,gc,gc,g

ic,gd ωω

θ=

ω−ω (6b)

In case of a lag compensator,

• the gain Kc has to be determined to satisfy the steady state requirement. • the desired gain crossover frequency ωg,c has to be selected such that

• θ < 0 or arg[G(jωg,c)H(jωg,c)] > –180° + φ m (7a) • ωg,c < ωg,u (7b) • Kc|G(jωg,c)H(jωg,c)| > 1 (7c) • cosθ < Kc|G(jωg,c)H(jωg,c)| (7d)

These requirements from (7a) through (7d) are to ensure that the angular contribution of the compensator will be negative, the peak angular contribution of the compensator will occur at frequencies lower than the gain crossover frequency of the uncompensated system, the magnitude contribution of the compensator will be less than its low frequency magnitude contribution, and the compensator itself as a subsystem will be stable, respectively. Note that if the requirement (7c) is satisfied, then the requirement (7d) is automatically satisfied.

In case of a lead compensator, • the gain Kc has to be determined to satisfy the steady state requirement. • The desired gain crossover frequency ωg,c has to be selected such that

• θ > 0 or arg[G(jωg,c)H(jωg,c)] < –180° + φ m (8a) • ωg,c > ωg,u (8b) • Kc|G(jωg,c)H(jωg,c)| < 1 (8c) • cosθ > Kc|G(jωg,c)H(jωg,c)| (8d)

These requirements from (8a) through (8d) are to ensure that the angular contribution of the compensator will be positive, the peak angular contribution of the compensator will occur at frequencies higher than the gain crossover frequency of the uncompensated system, the magnitude contribution of the compensator will be more than its low frequency magnitude contribution, and the compensator itself as a subsystem will be stable, respectively. Note that if the requirement (8d) is satisfied, then the requirement (8c) is automatically satisfied.

In case of a PID compensator, • the gain Ki has to be determined to satisfy the steady state requirement. • The desired gain crossover frequency ωg,c has to be selected such that

• –90° < θ < +90° (9a) • Ki|G(jωg,c)H(jωg,c)|+ωg,csinθ > 0 if θ < 0 (9b)

These requirements are to ensure that the necessary angular contribution can really be supplied by a PID controller and a positive value for Kd can be obtained, respectively.

ME 442 – Analytical Compensator Design in Frequency Domain Page

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In case of a PD compensator,

• the equation (6b) reduces to

)j(H)j(GsinK

c,gc,gc,gd ωω

θ=ω (10)

• therefore, in order to obtain a positive value for Kd, the desired gain crossover frequency ωg,c has to be selected such that

• θ > 0 or arg[G(jωg,c)H(jωg,c)] < –180° + φ m (11) In case of a PI compensator,

• the equation (6b) reduces to

)j(H)j(GsinK

c,gc,gc,g

i

ωωθ

=ω

− (12)

• therefore, in order to obtain a positive value for Ki, the desired gain crossover frequency ωg,c has to be selected such that

• θ < 0 or arg[G(jωg,c)H(jωg,c)] > –180° + φ m (13)